Arens–Fort space

In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr.

Definition

The Arens–Fort space is the topological space where is the set of ordered pairs of non-negative integers A subset is open, that is, belongs to if and only if:

does not contain or
contains and also all but a finite number of points of all but a finite number of columns, where a column is a set with fixed.

In other words, an open set is only "allowed" to contain if only a finite number of its columns contain significant gaps, where a gap in a column is significant if it omits an infinite number of points.

Properties

It is

Hausdorff
regular
normal

It is not:

second-countable
first-countable
metrizable
compact
sequential
Fréchet–Urysohn

There is no sequence in that converges to However, there is a sequence in such that is a cluster point of